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Question Number 20914 by Tinkutara last updated on 07/Sep/17

$$\mathrm{The}\mathrm{quadratic}\mathrm{equation}p\left(x\right)=0\mathrm{with}$$$$\mathrm{real}\mathrm{coefficients}\mathrm{has}\mathrm{purely}\mathrm{imaginary}$$$$\mathrm{roots}.\mathrm{Then}\mathrm{the}\mathrm{equation}p\left(p\left(x\right)\right)=0$$$$\mathrm{has}$$$$\left(1\right)\mathrm{Only}\mathrm{purely}\mathrm{imaginary}\mathrm{roots}$$$$\left(2\right)\mathrm{All}\mathrm{real}\mathrm{roots}$$$$\left(3\right)\mathrm{Two}\mathrm{real}\mathrm{and}\mathrm{two}\mathrm{purely}\mathrm{imaginary}$$$$\mathrm{roots}$$$$\left(4\right)\mathrm{Neither}\mathrm{real}\mathrm{nor}\mathrm{purely}\mathrm{imaginary}$$$$\mathrm{roots}$$

Answered by dioph last updated on 08/Sep/17

$$p\left(x\right)={ax}^2+c$$$$s\equiv \frac{\pm i\sqrt{4ac}}{2a}=\frac{\pm i\sqrt{ac}}{a}=\pm i\sqrt{\frac{c}{a}}$$$$p\left(p\left(x\right)\right)=a{({ax}^2+c)}^2+c$$$$=a(a^2{x}^4+2{acx}^2+c^2)+c$$$$=a^3{x}^4+2a^2{cx}^2+{ac}^2+c$$$$u\equiv x^2$$$$p\left(p\left(x\right)\right)=0\iff a^3{u}^2+2a^{2}cu+({ac}^2+c)=0$$$$\Rightarrow u=\frac{-2a^{2}c\pm \sqrt{4a^4{c}^2-4a^4{c}^2-4a^{3}c}}{2a^3}$$$$\Rightarrow u=\frac{-c\pm i\sqrt{c/a}}{a}=\frac{-c+s}{a}$$$$\Rightarrow x=\sqrt{u}\mathrm{is}\mathrm{neither}\mathrm{real}\mathrm{nor}\mathrm{purely}$$$$\mathrm{imaginary}$$

Commented by Tinkutara last updated on 08/Sep/17

$$\mathrm{Thank}\mathrm{you}\mathrm{very}\mathrm{much}\mathrm{Sir}!$$

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